Stress Intensity Factor (SIF) Solutions and Fatigue Crack Paths in Eccentric Circumferentially Cracked Round Bar (CCRB) in Tension

In this paper, a numerical modeling was developed to study (on the basis of the Paris law) the fatigue propagation paths of eccentric external (outer) cracks in circumferentially cracked round bars (CCRB) subjected to a cyclic type of loading in the form of either remote tensile loading or imposed axial displacement. Results show how the eccentricity (in relation to the wire axis) of the circular resistant ligament increases with the growth of outer circumferential cracks by subcritical fatigue mechanisms. This phenomenon is more pronounced when the solicitation consists of a remote tensile loading than when it is an axial displacement, when the initial eccentricity of the ligament increases (for a given initial diameter), and when the Paris exponent characteristic of the material rises. The paper also analyzes in depth the different situations regarding contact between crack faces during subcritical cyclic fatigue propagation, covering a wide range of cases including no contact, partial contact, and full contact depending on the ligament diameter (during the process of fatigue crack advance) and the relative eccentricity of the annular crack that loses its axial symmetry in relation to the round bar (cylinder) axis. In addition to the fatigue crack path study, closed-form stress intensity factor (SIF) solutions for the considered geometry (a cylinder with an outer annular crack) are provided in the form of third-degree polynomial expressions as a function of the ligament diameter and the crack eccentricity (both in dimensionless terms).


Introduction
In the case of metallic materials, circumferentially cracked round bar (CCRB) specimens, i.e., cylindrical specimens containing external (outer) circumferential cracks, are an interesting scientific alternative to the well-known standard specimens considered in the ASTM E399, Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness K Ic of Metallic Materials, cf. [1] for measuring the fracture toughness K IC of materials [2][3][4]. They have been successfully used to study cyclic fatigue crack propagation [5,6], stress corrosion cracking [7], and corrosion-fatigue [8].
From the experimental point of view, the main advantages of the aforesaid CCRB specimens are the low times for preparing the test specimens and the short duration of the experiments themselves [6], with the consequent saving of time. In addition, they exhibit fundamental scientific advantages in the purely conceptual framework, such as the achievement of a triaxial stress distribution in the vicinity of the crack tip, which makes a plane strain condition take place even for small specimen diameters [2], thereby providing a more reliable value of the fracture toughness of the material, more than a simply critical value of the stress intensity factor (SIF).
Nevertheless, the described CCRB specimens exhibit some disadvantages, such as their tendency to produce the appearance of certain specimen eccentricity when the crack grows [2], with the subsequent loss of axial symmetry of the specimen and potential problems of adequate experimental alignment of the sample with the grips of the testing machine. Such an undesirable phenomenon has been related to either the described nonsymmetric geometry of the sample, to the testing grips themselves, or even to the presence of a residual stress field in the sample or the non-uniform properties of the material [4].
In the scientific literature, there are SIF solutions in CCRBs subjected to axial tensile loading, containing either the more conventional symmetric situation [9][10][11] or the more realistic non-symmetric configuration with eccentric annular outer cracks [12,13]. In the latter case, additional bending stresses appear as a consequence of the lack of symmetry, and a new issue arises in the matter of possible contact between faces, with a wide range of situations covering no contact, partial contact, and full contact.
The sample eccentricity rises when the crack growths by fatigue (cyclic) loading [14][15][16] with the associated increasing loss of axial symmetry. In addition, the increase in initial eccentricity or the initial misalignment in the specimen produces a higher increase in eccentricity during fatigue propagation [15,16], thereby diminishing the time required for reaching the critical situation.
This paper provides closed-form SIF solutions for CCRBs in the form of third-degree polynomial expressions as a function of the ligament diameter and the crack eccentricity (both in dimensionless terms), covering a wide range of geometries and the two loading schemes considered in the present paper, namely remote tensile loading and imposed axial displacement.
In addition, the present paper analyzes the fatigue crack path evolution in a CCRB subjected to either remote cyclic tensile loading or imposed cyclic axial displacement, going further in the brief analysis initiated recently in a paper included in conference proceedings [17].

Numerical Modeling
Numerical modeling was performed to study the crack tip stress and strain fields and the fatigue crack propagation paths associated with external annular cracks in round bars subjected to imposed axial displacement (Figure 1a) or remote (axial) tensile loading ( Figure 1b). The specimen length is equal to twenty times its diameter D.
grows [2], with the subsequent loss of axial symmetry of the specimen and potentia lems of adequate experimental alignment of the sample with the grips of the testi chine. Such an undesirable phenomenon has been related to either the describe symmetric geometry of the sample, to the testing grips themselves, or even to the pr of a residual stress field in the sample or the non-uniform properties of the materi In the scientific literature, there are SIF solutions in CCRBs subjected to axial loading, containing either the more conventional symmetric situation [9][10][11] or th realistic non-symmetric configuration with eccentric annular outer cracks [12,13] latter case, additional bending stresses appear as a consequence of the lack of sym and a new issue arises in the matter of possible contact between faces, with a wide of situations covering no contact, partial contact, and full contact.
The sample eccentricity rises when the crack growths by fatigue (cyclic) loadi 16] with the associated increasing loss of axial symmetry. In addition, the increase in eccentricity or the initial misalignment in the specimen produces a higher increas centricity during fatigue propagation [15,16], thereby diminishing the time requi reaching the critical situation.
This paper provides closed-form SIF solutions for CCRBs in the form of thirdpolynomial expressions as a function of the ligament diameter and the crack ecce (both in dimensionless terms), covering a wide range of geometries and the two l schemes considered in the present paper, namely remote tensile loading and impos displacement.
In addition, the present paper analyzes the fatigue crack path evolution in a subjected to either remote cyclic tensile loading or imposed cyclic axial displaceme ing further in the brief analysis initiated recently in a paper included in conferen ceedings [17].

Numerical Modeling
Numerical modeling was performed to study the crack tip stress and strain fie the fatigue crack propagation paths associated with external annular cracks in roun subjected to imposed axial displacement ( Figure 1a) or remote (axial) tensile loadin ure 1b). The specimen length is equal to twenty times its diameter D.   The eccentric CCRB geometry with an annular external crack ( Figure 2) was characterized by the following parameters: round bar diameter D, maximum crack depth a max (related to point A), minimum crack depth a min (linked to point B), ligament diameter d, and ligament eccentricity ε (or, in other words, crack eccentricity, since the loss of ax-ial symmetry applies to both the internal circular ligament and to the external annular circumferential crack), i.e., The finite element method (FEM) analysis, performed with MSC. Marc 2 ware, has been used to simulate the eccentric CCRB subjected to tension (for impo displacement and for remote tensile loading), as explained in detail in a previou [13]. Only a quarter of the bar was modeled (taking advantage of the problem sym and 20-node isoparametric hexahedral elements were used, with the nodes close crack front shifted to the quarter-point position to model the crack tip singularity vergence study of the mesh size was performed, decreasing the size of the elemen area close to the crack front, and the SIF KI was obtained from the J-integral ca under plane strain conditions.
The basic hypothesis of the modeling is that the crack advances by fatigue cyc ing while the crack front maintains a circular shape, following the Paris-Erdog [18]: where C and m are the Paris parameters (characteristics of the material). In the ca of the fatigue crack path, carried out in successive iterations, only points A and circular crack front were taken into account. The advance ΔaA of the point A is the same throughout the calculation, w progress ΔaB of the point B is performed with the Paris law as: where KI * is the dimensionless SIF:

Stress Intensity Factor (SIF)
The dimensionless SIF KI * associated with points A and B of the crack front The finite element method (FEM) analysis, performed with MSC. Marc 2012 software, has been used to simulate the eccentric CCRB subjected to tension (for imposed axial displacement and for remote tensile loading), as explained in detail in a previous paper [13]. Only a quarter of the bar was modeled (taking advantage of the problem symmetry), and 20-node isoparametric hexahedral elements were used, with the nodes closest to the crack front shifted to the quarter-point position to model the crack tip singularity. A convergence study of the mesh size was performed, decreasing the size of the elements in the area close to the crack front, and the SIF K I was obtained from the J-integral calculation under plane strain conditions.
The basic hypothesis of the modeling is that the crack advances by fatigue cyclic loading while the crack front maintains a circular shape, following the Paris-Erdogan Law [18]: where C and m are the Paris parameters (characteristics of the material). In the calculation of the fatigue crack path, carried out in successive iterations, only points A and B at the circular crack front were taken into account. The advance ∆a A of the point A is the same throughout the calculation, while the progress ∆a B of the point B is performed with the Paris law as: where K I * is the dimensionless SIF:

Stress Intensity Factor (SIF)
The dimensionless SIF K I * associated with points A and B of the crack front was calculated as described above. It is seen that a magnification of SIF values is achieved at point A (K * IA) in the case of remote tensile loading (when compared to imposed axial displacement), mainly for the highest eccentricity value (ε/D = 0.30). The cause is the bending effect that appears when the axial loading is in remote tension, i.e., in the absence of displacement constraint, thereby allowing the rotation of the sample ends, which is favoured by the geometric eccentricity of the resistant ligament in relation to the bar axis. On the other hand, such a rotation is prevented in the case of imposed axial displacement, since the latter implies a constraint in the vicinity of the sample ends, a region in which rotation is impossible, thus reducing the bending effect and therefore diminishing the SIF value at point A (K * IA) when compared to the case of remote tensile loading, in which the aforesaid bending effect is quite pronounced. It is seen that a magnification of SIF values is achieved at point A (K * IA ) in the case of remote tensile loading (when compared to imposed axial displacement), mainly for the highest eccentricity value (ε/D = 0.30). The cause is the bending effect that appears when the axial loading is in remote tension, i.e., in the absence of displacement constraint, thereby allowing the rotation of the sample ends, which is favoured by the geometric eccentricity of the resistant ligament in relation to the bar axis. On the other hand, such a rotation is prevented in the case of imposed axial displacement, since the latter implies a constraint in the vicinity of the sample ends, a region in which rotation is impossible, thus reducing the bending effect and therefore diminishing the SIF value at point A (K * IA ) when compared to the case of remote tensile loading, in which the aforesaid bending effect is quite pronounced.
For lower levels of eccentricity (for all the values of the specimen ligament diameter), no contact between crack faces appears, and the situation remains closer to the symmetric case where there is no dissimilarity between the crack-tip stress-strain fields in the vicinity of point A and in the near-tip area of point B (see Figure 2). This happens for both loading scheme situations considered in the present paper, namely, imposed axial displacement and remote tensile loading.  For lower levels of eccentricity (for all the values of the specimen ligament diameter), no contact between crack faces appears, and the situation remains closer to the symmetric case where there is no dissimilarity between the crack-tip stress-strain fields in the vicinity of point A and in the near-tip area of point B (see Figure 2). This happens for both loading scheme situations considered in the present paper, namely, imposed axial displacement and remote tensile loading.
For higher levels of eccentricity (for small values of the specimen ligament diameter), contact arises between crack faces in the vicinity of point B (see Figure 2) as a consequence of the bending effect appearing in the specimen, and the situations differ more from the symmetric case in such a manner that there is a certain dissimilarity between the crack-tip stress-strain fields in the vicinity of point A and in the near-tip area of point B (see Figure  2). The described phenomenon of contact appears for both loading scheme situations considered in the present paper, namely, imposed axial displacement and remote tensile loading, and it takes place in two sub-forms: (i) partial contact in the cases of values of relative ligament diameter d/D ≤ 0.6 and from certain eccentricity level ε/D (this level being higher as d/D increases, following a For higher levels of eccentricity (for small values of the specimen ligament diameter), contact arises between crack faces in the vicinity of point B (see Figure 2) as a consequence of the bending effect appearing in the specimen, and the situations differ more from the symmetric case in such a manner that there is a certain dissimilarity between the cracktip stress-strain fields in the vicinity of point A and in the near-tip area of point B (see Figure 2). The described phenomenon of contact appears for both loading scheme situations considered in the present paper, namely, imposed axial displacement and remote tensile loading, and it takes place in two sub-forms: (i) partial contact in the cases of values of relative ligament diameter d/D ≤ 0.6 and from certain eccentricity level ε/D (this level being higher as d/D increases, following a more or less quasi-linear relationship, as shown in Figure 5 in the matter of partial contact beginning).
being lower as d/D increases, following curvilinear relationship tending to a ho asymptotic level, as shown in Figure 5 in the matter of full contact beginning).
It is important to emphasize that in this case of full contact, the dissimilarity b the crack-tip stress-strain fields is maximum since the maximum SIF is achieved A (very intense near-tip field as a consequence of specimen bending), whereas at posite point B the near-tip field is null, and the SIF is zero. With regard to the differences between both loading scheme situations consi the present paper, namely, imposed axial displacement and remote tensile load in the matter of the phenomenon of contact, it is possible to say that the event of is more likely (and more pronounced) in the case of remote tensile loading tha situation of imposed axial displacement, since the bending effect has a more im role in the former than in the latter, i.e., an axial displacement, in a certain sense, p (or at least, diminishes) the bending effect, thereby requiring higher levels of ecc to generate both partial and full contact (when compared to the case of remote loading, which enhances bending and promotes both partial and full contact e lower levels of specimen eccentricity).
The numerical results with regard to the dimensionless SIF KI * were fitted to mial expressions of the third degree using a least squares method as a functio ligament diameter d/D and the crack eccentricity ε/D (both in dimensionless term ering a wide range of geometries, thereby providing a high level of generality to th since such analytical expressions are applicable to any fracture mechanics, dama ance, and structural integrity analysis of CCRB. The closed-form dimensionless SIF solutions for the different cases (imposed placements and remote tensile loading; no contact and partial contact) are given at bo front points A and B as follows: • For imposed axial displacement and no contact: (ii) full contact in the cases of values of relative ligament diameter d/D ≤ 0.6 (the same interval as in the case of a partial contact) and from certain eccentricity level ε/D (this level being lower as d/D increases, following curvilinear relationship tending to a horizontal asymptotic level, as shown in Figure 5 in the matter of full contact beginning).
It is important to emphasize that in this case of full contact, the dissimilarity between the crack-tip stress-strain fields is maximum since the maximum SIF is achieved at point A (very intense near-tip field as a consequence of specimen bending), whereas at the opposite point B the near-tip field is null, and the SIF is zero.
With regard to the differences between both loading scheme situations considered in the present paper, namely, imposed axial displacement and remote tensile loading, and in the matter of the phenomenon of contact, it is possible to say that the event of contact is more likely (and more pronounced) in the case of remote tensile loading than in the situation of imposed axial displacement, since the bending effect has a more important role in the former than in the latter, i.e., an axial displacement, in a certain sense, prevents (or at least, diminishes) the bending effect, thereby requiring higher levels of eccentricity to generate both partial and full contact (when compared to the case of remote tensile loading, which enhances bending and promotes both partial and full contact even for lower levels of specimen eccentricity).
The numerical results with regard to the dimensionless SIF K I * were fitted to polynomial expressions of the third degree using a least squares method as a function of the ligament diameter d/D and the crack eccentricity ε/D (both in dimensionless terms), covering a wide range of geometries, thereby providing a high level of generality to the paper since such analytical expressions are applicable to any fracture mechanics, damage tolerance, and structural integrity analysis of CCRB.
The closed-form dimensionless SIF solutions for the different cases (imposed axial displacements and remote tensile loading; no contact and partial contact) are given at both crack front points A and B as follows:

•
For imposed axial displacement and no contact:  The six plots in Figures 6 and 7 show a general trend of decreasing relationship between the ligament eccentricity and the ligament diameter (it must be taken into account that the horizontal axes of the six plots in Figures 6 and 7 are expressed in decreasing units). Therefore, as the ring-shaped outer crack grows towards the inner region of the cylinder, the ligament diameter decreases, whereas the ligament eccentricity (that is a measure of the full specimen eccentricity) increases, i.e., the specimen deviates from the axisymmetric case.

Fatigue Crack Propagation Paths
Another general trend of the six plots in Figures 6 and 7 is the shape of the curves. All plots exhibit a curvilinear aspect at the beginning (low sample eccentricities and high ligament diameters), whereas after this initial period, all plots tend to a quasi-linear shape, i.e., the quasi-linear relationship between eccentricity and ligament diameter for the final stages of fatigue crack growth, i.e., for the highest levels of eccentricity itself.
The grey dashed lines in the six plots in Figures 6 and 7 correspond to the beginning of the appearance of partial contact or full contact between the crack faces ( Figure 5), a phenomenon occurring for lower values of relative eccentricity in the case of remote tensile loading than in the situation of imposed axial displacement, due to a bending effect in the sample in the first case, as discussed in previous sections of the present paper and in reference [13]. Figures 8 and 9 show the fatigue crack paths in the form of circular crack front evolution for given relative diameters of the ligament (from an initial value up to 0.30 with increments of 0.05) during fatigue crack growth from the initial geometries with two initial ligament diameters (d/D) 0 = {0.75, 0.60} and two initial ligament eccentricities (ε/D) 0 = {0.0025, 0.0100}. The crack fronts in blue (m = 2) and brown (m = 4) indicate that there is no contact between the crack faces, whereas the crack fronts in pink (m = 2) and red (m = 4) indicate that there is partial contact between the faces of the crack. Finally, the crack fronts drawn in dark grey colours indicate that full contact takes place between the crack faces.   The crack growth caused by fatigue obliges the relative eccentricity of the ligament to increase when the net diameter of the specimen (i.e., the ligament diameter) diminishes during the propagation phase. In addition, the values of the relative eccentricity during the crack advance are higher in the following three situations: (i) When the external action over the cylindrical specimen (round bar containing a ringshaped external circumferential crack) is a remote tensile loading instead of an imposed axial displacement (in the latter case, boundary conditions prevent sample bending).
(ii) With the increase in the Paris exponent m characteristic of the material behaviour, i.e., the higher the exponent m of the Paris Law (linked with a more accelerated fatigue crack growth), the higher the specimen eccentricity.
(iii) When the initial relative eccentricity of the ligament increases (for a given initial relative diameter), i.e., an elevated initial eccentricity promotes the further increase in the eccentricity of the sample even more.
The curves ε/D-d/D for different initial relative eccentricities of the ligament, with a given initial relative diameter, are closer to each other with the increase in the initial relative diameter of the ligament and when the initial relative eccentricities are more elevated. In addition, in the case of high initial relative diameters of the ligament, (d/D) 0 = 0.75, the curves for different initial eccentricities become closer when the external action consists of remote tensile loading. On the other hand, in the opposite case of low initial relative diameters of the ligament, (d/D) 0 = 0.45, such curves become closer when the external action consists of imposed axial displacement.
During fatigue crack propagation, all curves representing specimen eccentricity (or ligament eccentricity) as a function of ligament diameter, i.e., plots ε/D versus d/D tend towards quasi-straight lines that are, in addition, quasi-parallel between them. On assuming a strictly linear shape, all plots are strictly parallel, thereby sharing a common slope for a given material, i.e., a unique slope for a specific Paris exponent m, and different for distinct materials (distinct Paris m exponent).
The aforesaid equal slope of the ε/D-d/D plots corresponds to the growth when point B at the crack front remains immobile (the instant from which the supposition of a maintained circular crack front is impossible). Nevertheless, it is from the beginning of partial contact (or even before for the materials with the most elevated Paris m exponents) that the curves ε/D-d/D become straight lines, as observed in Figures 6 and 7.
The present results are fully general for any CCRB specimen since a wide range of cracked geometries with very different initial depths of the external (outer) ring-shaped circumferential crack in the round bar were considered. With regard to the material, numerical results are only dependent on the Paris exponent m of the material. As a range m = {2,3,4} was considered, the results of the present paper are fully applicable to a really wide range of metallic materials.
The results are also applicable to environmentally assisted fatigue or corrosion-fatigue processes of metals and alloys by using a modified exponent of the Paris law for the situation of corrosion-fatigue, i.e., affected by the presence of a harsh or aggressive environment promoting an accelerated fatigue crack growth rate.

Conclusions
The following conclusions may be drawn regarding fatigue crack propagation paths in eccentric circumferentially cracked round bar (CCRB) specimens under tensile cyclic (fatigue) loading with remote tensile loading and imposed axial displacements: (i) Closed-form stress intensity factor (SIF) solutions are provided for an eccentric outer (external) crack located in circumferentially cracked round bars (CCRB) under axial tensile loading in the form of remote tensile loading or imposed axial displacement. (ii) Such SIF solutions are provided in the form of third-degree polynomial expressions as a function of the ligament diameter and the specimen eccentricity (both in dimensionless terms), covering a wide range of CCRB geometries. (iii) The fatigue propagation paths associated with eccentric circumferential cracks located in round bars under tensile cyclic loading produce a clear augmentation of the eccentricity of the resistant ligament of the bar. (iv) The described phenomenon is more intense when there is an increase in the initial relative eccentricity (for the same given initial relative diameter) and in the Paris exponent m characteristic of the material. (v) In the eccentric CCRBs in which a remote tensile loading is applied, the increase in specimen eccentricity is higher than in those bars subjected to imposed axial displacement due to a bending effect. (vi) The reason for the aforesaid higher increase in specimen eccentricity in the case of remote tensile loading is the external constraint (boundary condition) created by the imposed axial displacement, thereby preventing specimen bending. (vii) The plots' relative eccentricity versus relative diameter (ε/D-d/D) are straight lines of equal slope from the beginning of partial contact, or even before for the materials with more elevated exponent m of the Paris law. (viii) Results possess a high level of generality: the provided closed-form SIF solutions (analytical expressions) are applicable to any fracture mechanics, damage tolerance, and structural integrity analyses in which CCRB geometries are involved.